A144207 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges where each maximally connected subgraph consists of a single node or has a unique cycle of length 3.

1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 4, 12, 1, 0, 0, 10, 60, 150, 1, 0, 0, 20, 180, 900, 2160, 1, 0, 0, 35, 420, 3150, 15180, 36015, 1, 0, 0, 56, 840, 8400, 60750, 291060, 688128, 1, 0, 0, 84, 1512, 18900, 182270, 1311240, 6300672, 14880348, 1, 0, 0, 120, 2520, 37800

Offset

0,14

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

T(n,0) = 1, T(n,k) = 0 if k<0 or n<k, else T(n,k) = C(n-1,2)*n^(n-3) if k=n, else T(n,k) = T(n-1,k) + Sum_{j=2..k-1} C(n-1,j) T(j+1,j+1) T(n-1-j,k-j-1).

Example

T(5,4) = 60 = 5*12, because there are 5 possibilities for a single node and T(4,4) = 12:
.1-2. .1-2. .1-2. .1.2. .1.2. .1-2. .1.2. .1.2. .1-2. .1-2. .1-2. .1-2.
.|X.. .|/|. .|/.. ..X|. .|/|. ../|. .|X.. .|\|. .|\.. ..X|. .|\|. ..\|.
.3.4. .3.4. .3-4. .3-4. .3-4. .3-4. .3-4. .3-4. .3-4. .3.4. .3.4. .3-4.
Triangle begins:
1;
1, 0;
1, 0, 0;
1, 0, 0, 1;
1, 0, 0, 4, 12;
1, 0, 0, 10, 60, 150;

Maple

T:= proc(n, k) option remember; if k=0 then 1 elif k<0 or n<k then 0 elif k=n then binomial(n-1, 2) *n^(n-3) else T(n-1, k) +add(binomial(n-1, j) * T(j+1, j+1) *T(n-1-j, k-j-1), j=2..k-1) fi end: seq(seq(T(n, k), k=0..n), n=0..11);

Mathematica

t[n_, k_] := t[n, k] = Which[k == 0, 1, k < 0 || n < k, 0, k == n, Binomial[n-1, 2]*n^(n-3), True, t[n-1, k] + Sum[Binomial[n-1, j]*t[j+1, j+1]*t[n-1-j, k-j-1], {j, 2, k-1}]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 11}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

Crossrefs

Columns 0, 1+2, 3, 4 give: A000012, A000004, A000292, A033486 or A112415. Diagonal gives: A053507. Row sums give: A144208. Cf. A007318.

Sequence in context: A350259 A330989 A073902 * A016487 A370707 A104063

Adjacent sequences: A144204 A144205 A144206 * A144208 A144209 A144210

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 14 2008

Status

approved